Intro to Multivariable Calculus

Finding Extrema

The surface area of the box \( A \) is dependent on the base width \( x \) and the height \(y.\)

We express this relationship explicitly as
\[ A(x,y) = x^2 + 4 x y.\] Writing the surface area as \( A (x,y) \) tells us that it is a function of both base width and height.

A company wants to produce such a box with base width at least 4 units and a height of at least 1 unit. If box material is 4 dollars per unit area, what's the cost of the cheapest box that can be produced?

Hint: This problem actually doesn't require any calculus, just some intuition.

Finding Extrema

Intro to Multivariable Calculus

Finding Extrema

By introducing partial derivatives \( \frac{\partial f}{\partial x }\) and \(\frac{\partial f}{\partial y},\) we took our first steps into the larger world of multivariable optimization. We'll have a whole chapter on these derivatives, but we first need to know more about functions and their graphs.

Graphs are visual tools that can help us in our quest for extrema. Take a look at the animations below. The surface represents the graph of \( f(x,y) = 2 x y e^{1-x^2-y^2} \) on the larger square \( -2 \leq x, y \leq 2.\) The mountain tops correspond to maxima, and the pits represent minima.

Surface graphs can also help us begin to understand one of the other major pillars of multivariable calculus: integration. To graph surfaces, we need a coordinate system in space, which is the topic of our next unit. With this detour out of the way, we'll come back to many-variable integration to finish off our chapter.